![Page 1: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/1.jpg)
http://www.emceoc.org
Understanding the Law of Propagation
of UncertaintiesEmma Woolliams
4 April 2017
![Page 2: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/2.jpg)
The Law of Propagation of
Uncertainties
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
![Page 3: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/3.jpg)
Things you might already “know”
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Has a sensitivity coefficient
Adding in quadrature (% or units)
Averages reduce by 1 n
This term is to do
with correlation
![Page 4: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/4.jpg)
Building on that starting point
• The Law of Propagation of Uncertainties
• How many readings you should average
• What correlation is
Has a sensitivity coefficient
Adding in quadrature (% or units)
Averages reduce by
This term is to do with correlation
1 n
2c
2
2
1
1
1 1
2 ,
n
iii
n n
i ji ji j i
u y
fu x
x
f fu x x
x x
![Page 5: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/5.jpg)
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
![Page 6: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/6.jpg)
THE LAW OF PROPAGATION
OF UNCERTAINTIES
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
![Page 7: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/7.jpg)
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
![Page 8: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/8.jpg)
Sensitivity Coefficients
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
![Page 9: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/9.jpg)
How sensitive is
my result to
this?
![Page 10: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/10.jpg)
How sensitive is
my result to
this?
Do an
experiment
to find out!
![Page 11: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/11.jpg)
Lamp power supply:
What is the sensitivity of the irradiance to the lamp current?
![Page 12: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/12.jpg)
0.985
0.99
0.995
1
1.005
1.01
1.015
250 500 750 1000 1250 1500 1750 2000 2250 2500
Sig
nal ra
tio
wavelength / nm
Ratio 27.21 A / 27.19 A
Measured ratio0.2 A10 %
![Page 13: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/13.jpg)
PTFE Sphere phase transition
6 °C
1.5 %
![Page 14: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/14.jpg)
How sensitive is
my result to
this?
Calculate it from
the
measurement
equation
![Page 15: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/15.jpg)
Analytical sensitivity coefficients
ixi
fc
x
1 2 3, , ,y f x x x
The local slope:
Translating a change in one
parameter to a change in
the other
![Page 16: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/16.jpg)
When it works well
S
light
1V
V
S light darkV V V
S
dark
1V
V
Very simple case
![Page 17: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/17.jpg)
When it works well
2FEL 0 45 cal
s 2use
E dL
d
2
s FEL 0 45 cal s
3
use use use
22
L E d L
d d d
Inverse square law leads to a 2
![Page 18: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/18.jpg)
When it doesn’t work well
• Because you can’t write a relationship
• Because it’s too difficult to differentiate
• Because it’s a program not an equation
![Page 19: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/19.jpg)
How sensitive is
my result to
this?
Model
it!
![Page 20: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/20.jpg)
Radiance through double aperture
system
2 2
1 2-d 2
2 2 2 2 2 2 2 2
1 2 1 2 1 2
2
4
r rg
r r x r r x r r
1A g L
![Page 21: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/21.jpg)
By differentiating …
22 2 2 2 2
1 2 1 2
2 2 2
1 2
4r r d r r
r r d
2 2
1 2-d 2
2 2 2 2 2 2 2 2
1 2 1 2 1 2
2
4
r rg
r r x r r x r r
2 2 1
1 22g r r
2 2 2
1 2 1 2
1
2 2 2
1 2 2 1
2
2 2
1 2
4 21
4 21
4
r r r rg
r
r r r rg
r
r r dg
d
![Page 22: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/22.jpg)
Or by “modelling”…
![Page 23: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/23.jpg)
Sensitivity to atmospheric model choice
Tuz Gölü CEOS comparison
2010
Difference between continental
and desert model lead up to 3%
difference in 𝜌 values
![Page 24: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/24.jpg)
Monte Carlo simulation
![Page 25: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/25.jpg)
Sensitivity Coefficients
How sensitive is
my result to
this?
Experimentally
Mathematically
Numerically
ixi
fc
x
Vary in the lab and see
what happens
Vary in the
model and
see what
happens
All are acceptable
![Page 26: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/26.jpg)
Adding in Quadrature
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
![Page 27: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/27.jpg)
Why quadrature?
Uncertainty Error
Describes the spread Difference to the
(unknowable) true
value
It’s very unlikely
that all will be
maximum in
same direction!
Combining
distributions
leads to the
quadrature rule
![Page 28: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/28.jpg)
Uncertainty expression
Relative uncertainty:
5 mW m-2 nm-1 ± 0.2 %
i.e. uncertainty expressed as a percentage
Absolute uncertainty:
5 mW m-2 nm-1 ± 0.01 mW m-2 nm-1 i.e.
uncertainty expressed in the native
measurement units
![Page 29: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/29.jpg)
Uncertainty expression
Relative uncertainty:
5 mW m-2 nm-1 ± 0.2 %
i.e. uncertainty expressed as a percentage
Absolute uncertainty:
5 mW m-2 nm-1 ± 0.01 mW m-2 nm-1 i.e.
uncertainty expressed in the native
measurement units
![Page 30: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/30.jpg)
Adding in quadrature for absolute
models
y BA C 1B
y y
CA
y
2 2 2 2Au y u u uB C
![Page 31: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/31.jpg)
Adding in quadrature for absolute
models
y BA
1A
y
22 2 21Au u By u
1B
y
2 2 2Au y u Bu
![Page 32: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/32.jpg)
What behaves like an absolute model?
Distance = Measured + aperture thickness
Signal = Light signal – Dark signal
![Page 33: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/33.jpg)
Uncertainty expression
Relative uncertainty:
5 mW m-2 nm-1 ± 0.2 %
i.e. uncertainty expressed as a percentage
Absolute uncertainty:
5 mW m-2 nm-1 ± 0.01 mW m-2 nm-1 i.e.
uncertainty expressed in the native
measurement units
![Page 34: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/34.jpg)
Adding in quadrature for relative
modelsy BA
y y
A AB
2 2 2
u y u
Ay
B
B
A u
y y
B BA
2 2
2 2 2y yu y BA
BAu u
![Page 35: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/35.jpg)
Adding in quadrature for relative
modelsy BA
1
BA A
y y
2 2 2
u y u
Ay
B
B
A u
2
y A y
BB B
2 2
2 2 2Ay y
BB
u uA
y u
![Page 36: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/36.jpg)
Adding in quadrature for relative
models
2y A B
2
1y y
BA A
2 2 2
2u y u A B
BA
u
y
3
2 2y A y
B BB
2 2
2 2 22A
y yB
By u u
Au
![Page 37: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/37.jpg)
What behaves like a relative model?
• Most radiometric equations!
• Anything where uncertainties are in %
Gain = Radiance / Signal
2FEL 0 :45 cal
s 2use
E dL
d
s s
FEL FEL
L L
E E
s s
0 :45 0 :45
L L
s s
use use
2L L
d d
2 2 22
s 0 :45 useFEL
s FEL 0 :45 use
2u L u u du E
L E d
![Page 38: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/38.jpg)
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
![Page 39: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/39.jpg)
HOW MANY READINGS
SHOULD BE AVERAGED?
![Page 40: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/40.jpg)
At the end of this module, you should
understand
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• What the ‘other term’ – the correlation bit means
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
![Page 41: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/41.jpg)
Applying the Law of Propagation of
Uncertainties to an average
Averaging three readings
1 2 3M
3
E E EE
M M M
1 2 3
1
3
E E E
E E E
2 2 2
2 2 2 2
M 1 2 3
1 1 1
3 3 3u E u E u E u E
1 2 3 iu E u E u E u E
2
2 2
M
13
3iu E u E
2 2
M 2
3
3iu E u E
2 2
M
1
3iu E u E
2
2
M3
iu Eu E
M3
iu Eu E
![Page 42: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/42.jpg)
Applying the Law of Propagation of
Uncertainties to an average
Averaging n readings
M 1
n
iiE E n
M 1
i
E
E n
2 2
M 1
n
iiu E u E n
2
2 2
M
1iu E n u E
n
2 2
M
1iu E u E
n
2
2
M
iu Eu E
n
M
iu Eu E
n
![Page 43: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/43.jpg)
Average light – average dark
s light, dark,
1 1
1 1N M
i j
i i
V V VN M
dark,
dark,
1, 1, ,M
jV
j
Vc j
V M
light dark
2 2
2 V V
V
u uu
N M
light, dark,
2 2
2 2 2
1 1
1 1i j
N M
V V V
i j
u u uN M
light,
light,
1, 1, , N
iV
i
Vc i
V N
![Page 44: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/44.jpg)
Taking enough readings
![Page 45: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/45.jpg)
Applying the Law of Propagation of
Uncertainties to an average
Averaging three readings
1 2 3M
3
E E EE
M M M
1 2 3
1
3
E E E
E E E
2 2 2
2 2 2 2
M 1 2 3
1 1 1
3 3 3u E u E u E u E
1 2 3 iu E u E u E u E
2
2 2
M
13
3iu E u E
2 2
M 2
3
3iu E u E
2 2
M
1
3iu E u E
2
2
M3
iu Eu E
M3
iu Eu E
![Page 46: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/46.jpg)
What is the uncertainty associated
with each reading?
1 2 3 iu E u E u E u E
Uncertainty
Describes the spread
Standard
deviation?
![Page 47: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/47.jpg)
The dangers of taking a standard deviation
of a small number of readings
Reading numberValu
es a
nd c
um
ula
tive S
t d
ev
![Page 48: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/48.jpg)
The dangers of taking a standard deviation
from a small number of readings
![Page 49: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/49.jpg)
What you should do
Use more readings!
• Take more measurements today
M
iu Eu E
n
Better estimate of
uncertainty
More to average
![Page 50: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/50.jpg)
What you should do
Use more readings!
• Have a facility commissioning phase where you
take more measurements
M
iu Eu E
n
Better estimate of
uncertainty
Today’s!
From standard deviation of 10: iu E
Today two readings, 2n
![Page 51: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/51.jpg)
What you should do
Use more readings!
• Compare data from different days
Even if there is a drift:
e.g Use standard deviation of the
difference
between two readings
![Page 52: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/52.jpg)
What you should do
Use more readings!
• Compare data from different wavelengths
(smooth out the bumps)
![Page 53: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/53.jpg)
What you should do
Increase the estimate
• Increase standard uncertainty
Will be in revised GUM
2
light2
light,mean
1
3
sNu
N N
![Page 54: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/54.jpg)
What you should do
Use more readings!
• Take more measurements today
• Have a facility commissioning phase
• Compare data from different days
• Compare data from different wavelengths
Increase the estimate
• Increased standard uncertainty
![Page 55: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/55.jpg)
KNOWING WHEN TO STOP!
Use more readings!
But only when it’s worth it!
![Page 56: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/56.jpg)
The metre: 1791 - 1799
Pierre-
Françoise-André
Méchain
Jean-Baptiste-
Joseph
Delambre
The Measure of all Things - Ken Alder
![Page 57: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/57.jpg)
The uncertainties don’t keep dropping
forever
ADEV
Line Fit
Lower Bound
Upper Bound
AlaVar 5.2
Allan STD DEV
Produced by AlaVar 5.2
time / s 100 1000 10000
0.0001
0.001
![Page 58: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/58.jpg)
The Allan Deviation – software
ADEV
Lower Bound
Upper Bound
AlaVar 5.2
Allan STD DEV
Produced by AlaVar 5.2
time / s 100 1000
1E-5
0.0001
www.alamath.com
![Page 59: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/59.jpg)
At the end of this module, you should
understand
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
![Page 60: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/60.jpg)
CORRELATION
![Page 61: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/61.jpg)
At the end of this module, you should
understand
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
![Page 62: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/62.jpg)
The Law of Propagation of
Uncertainties
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
![Page 63: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/63.jpg)
Truei iE E S R
Correlation: Type A and Type B
methods
Correlation occurs when there is something in common between the
different quantities, or between the measured values being combined.
Type A: From the data Type B: From knowledge
This is where the
correlation comes
from!
Systematic
Effects!
![Page 64: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/64.jpg)
Correlation: Type A and Type B
methods
Correlation occurs when there is something in common between the
different quantities, or between the measured values being combined.
Type A: From the data
1
1,
1
ni i
i X Y
X X Y Yr x y
n s s
, ,u x y u x u y r x y
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
![Page 66: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/66.jpg)
Truei iE E S R
Correlation: Type A and Type B
methods
Correlation occurs when there is something in common between the
different parameters, or between the measurements being combined.
Type B: From knowledge
This is where the
correlation comes
from!
Systematic
Effects!
Calculate covariance
Remove covariance
![Page 67: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/67.jpg)
Averaging partially correlated data
Truei iE E S R 1 2 3M
3
E E EE
True 1 2 3M
3 3
3 3 3
E R R RSE
2
2 2
M3
iu Ru E u S
![Page 68: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/68.jpg)
Averaging Partially correlated data
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Truei iE E S R
2 2
i iu E u S u R
2, ;i ju E E u S i j
![Page 69: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/69.jpg)
Averaging partially correlated data
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
1 2 1 3 1 4
2 3 2 4
3 4
, , , , , ,
, , , ,
, ,
u E E u E E u E E
u E E u E E
u E E
![Page 70: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/70.jpg)
Averaging partially correlated data
Truei iE E S R 1 2 3
M3
E E EE
M M M
1 2 3
1
3
E E E
E E E
2, ;i ju E E u S i j
2 2
i iu E u S u R
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
2 2
2 2 2
M
1 13 3
3 3iu E u S u R
1 2 2 3
1 3 21 12
3 3u S
2
213 2
3u S
2 2
2 2 2
M
1 13 6 3
3 3iu E u S u R
2
2 2
M3
iu Ru E u S
![Page 71: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/71.jpg)
Systematic and random effects: Lamp
measured 5 times (continued)
Systematic effects Random effects
Reference calibration Noise
Alignment Lamp current fluctuation
Lamp current setting
Temperature sensitivities
,S u S ,i iR u R
Truei iE E S R
2 2 2
i iu E u S u R
2
2 2
M
iu Ru E u S
n
![Page 72: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/72.jpg)
Averaging partially correlated data
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
1 2 1 3 1 4
2 3 2 4
3 4
, , , , , ,
, , , ,
, ,
u E E u E E u E E
u E E u E E
u E E
![Page 73: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/73.jpg)
Covariance Matrix
21 1 2 1
22 1 2 2
21 2
, ,
, ,
, ,
1 2
1
2
n
E n
n n n
u E u E E u E E
u E E u E u E E
u E E u E E un E
n
U
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
1 2 1 3 1 4
2 3 2 4
3 4
, , , , , ,
, , , ,
, ,
u E E u E E u E E
u E E u E E
u E E
![Page 74: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/74.jpg)
Covariance Matrix Version
1 2y
n
f f f
x x x
C
21 1 2 1
22 1 2 2
21 2
, ,
, ,
, ,
1 2
1
2
n
E n
n n n
u E u E E u E E
u E E u E u E E
u E E u E E un E
n
U
Covariance matrix
with absolute
variance (squared
uncertainty) and
covariance
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
![Page 75: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/75.jpg)
Covariance matrix
21 1 2 1
22 1 2 2
21 2
, ,
, ,
, ,
1 2
1
2
n
E n
n n n
u E u E E u E E
u E E u E u E E
u E E u E E un E
n
U
T 1 1i i i iE E S R s r
2 2 2 2 2
,2 2
( )
( ).
i i i
E ij
i j
E u S u R u s u r i jU
E E u S u s i j
![Page 76: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/76.jpg)
Negative sensitivity coefficients
y BA 1A
y
1
B
y
22 2 21Au u By u
2 2 2Au y u Bu
E.g. Signal = Light - Dark
![Page 77: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/77.jpg)
Negative sensitivity coefficients
y BA 1A
y
1
B
y
2 2 2 2 1 1 ,Bu y u AA u u B
E.g. Signal = Light - Dark
2 2 2 2 ,u y u u AB BA u
Correlation
reduces
uncertainty
![Page 78: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/78.jpg)
Positive sensitivity coefficients
y BA 1A
y
1
B
y
2 2 2 2 1 1 ,BAu y u u u A B
E.g. Distance = measured + thickness
2 2 2 2 ,u y u u AB BA u
Correlation
increases
uncertainty
![Page 79: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/79.jpg)
But I don’t know what the covariance is!
• Think of the worst-case scenario
Does correlation increase or decrease the
uncertainty?
Treat as “all random” and “all systematic”
![Page 80: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/80.jpg)
At the end of this module, you should
understand
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
![Page 81: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/81.jpg)
CONCLUSIONS AND
SUMMARY
![Page 82: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/82.jpg)
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
![Page 83: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/83.jpg)
Sensitivity Coefficients
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
How sensitive is
my result to
this?
Experimentally
Mathematically
Numerically
![Page 84: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/84.jpg)
Taking enough readings, but not too
many
M
iu Eu E
n
Better estimate of
uncertainty
More to average
ADEV
Line Fit
Lower Bound
Upper Bound
AlaVar 5.2
Allan STD DEV
Produced by AlaVar 5.2
time / s 100 1000 10000
0.0001
0.001
Allan Deviation
![Page 85: Understanding the Law of Propagation of Uncertainties · Uncertainty expression Relative uncertainty: 5 mW m-2 nm-1 ±0.2 % i.e. uncertainty expressed as a percentage Absolute uncertainty:](https://reader035.vdocuments.site/reader035/viewer/2022071016/5fcf0098bdf96f409711f734/html5/thumbnails/85.jpg)
Correlation
Truei iE E S R
Correlation occurs when there is something in common between the
different parameters, or between the measurements being combined.
Type B: From knowledge
This is where the
correlation comes
from!
Systematic
Effects!
Calculate covariance
Remove covariance